Analysis of a Discrete-time Queue with Modified Batch Service Policy and Batch-size-dependent Service

We consider a discrete-time single server batch service queue, with a modified bulk service rule, where jobs will process in batches with a minimum batch size of L and maximum batch size K . In addition, we let jobs join the service if the server has started the service and has less than K jobs. The service times of the batches depend on the size of the service batch. We find explicit analytical expressions for the joint and marginal probability distribution of the number of jobs in the system, in the queue, and with the server. We also compute several performance measures and illustrate their dependence on various system parameters. Finally, we match the presented results with some results in the literature and establish a link to the continuous-time counterpart.